**Embedded Fluid Structure Interaction Technique for Thin Deformable Structure**

**STATE OF THE ART**

**Masoud Davari**

Advisor: Dr. Pooyan Dadvand

Co-Advisor: Dr. Riccardo Rossi

Advisor: Dr. Pooyan Dadvand

Co-Advisor: Dr. Riccardo Rossi

**INTRODUCTION AND MOTIVATION**

**FSI**

Interaction of movable, rigid or deformable structures with internal or surrounding fluid flow

Influence of fluid and structure dynamics to each other:

Structure deform under the action of the fluid stresses

This deformation, in turn, changes the boundary conditions of the fluid flow (Fluid follows the structure displacement )

significant challenge:

large movement, or the large deformation of structure due to serve fluid stresses

geometrical complexity

Re-mesh processing is needed

expensive

difficulty in mapping the state variables from the old mesh to the new mesh

The approaches to the field of Fluid Structure Interaction for handling moving interfaces of an elastic solid structure can be subdivided into two classes,

conforming mesh

meshes are conformed to the interface where the physical boundary conditions are imposed. In this class due to the large deformation and/or movement of the solid structure, re-meshing is needed as the solution is advanced

non-conforming mesh

meshes are not conformed to the interface. As a result, the fluid and solid equations can be conveniently solved independently from each other with their respective grids, and re-meshing is not necessary.

conforming mesh methods

Arbitrary Lagrangian Eulerian method (ALE)

An ALE method allows arbitrary motion of grid/mesh points with respect to their frame of reference by taking the convection of these points into account

This method goes back to early works like (Belytschko et al. 1980; Belytschko and Kennedy 1978; CEL 1964; Codina et al. 2009; Hirt et al. 1974; J. Donea et al. 1977; J. Hughes et al. 1981)

conforming mesh methods

Particle Finite Element Method (PFEM)

The key feature of the PFEM is that the equations of motion of the fluid domain are formulated in a Lagrangian framework. In other word PFEM is a particular class of Lagrangian formulation aiming to solve problems involving the interaction between fluids and solids a unified manner.

The PFEM method for the solution of FSI problems is recently used by (Aubry et al. 2004; A Larese, R Rossi 2008; Oñate 2004; Oñate et al. 2003 ; Oñate and Periaux 2006 ).

advantage:

The convective term in the momentum conservation equation disappears.

disadvantage:

Re-meshing at each time step is required.

non-conforming mesh methods

Immersed Boundary Method

- A finite difference grid for the fluid domain

- An immersed set of non-conforming boundary points that are mutually interconnected

by an elastic law

- Solid boundary interacts with the fluid by means of local body forces applied to

the fluid at the position of the structure

- IB has been successfully applied in many application fields (Dillon and Fauci

2000; Gilmanov and Sotiropoulos 2005; Jung 1999; Zhu and Peskin 2002)

Fictitious Domain Method

non-conforming mesh methods

- Evolved from the field of finite elements

- Coupling is established by constraining fluid and rigid body at the interface using a

(distributed) Lagrange multiplier

- Lagrange multipliers consist of adding new equations to the global system of

equations that enforce the boundary conditions

- New unknowns (the Lagrange multipliers) need also to be added to the problem

advantage:

mesh updating not required

disadvantage:

the system of equations becomes larger, and the space for the Lagrange multipliers has to be carefully chosen so that the final formulation is stable

non-conforming mesh methods

Extended Immersed Boundary Method

and

Immersed Finite Element Method

- describe the solid using the finite element method

- describe the fluid is formulated using a finite difference or finite element method

- The coupling is performed using the discrete dirac delta functions that find

their origin in the mesh-less Reproducing Kernel Particle Method (RKPM).

- Within these methods, the conservative and computationally stable coupling of solid and

fluid can be realized by higher-order schemes, increasing the numerical effort.

In these method that mentioned in non-conforming mesh class ,the interaction of the fluid and the solid is taken into account through a force term which appears either in the strong or in the weak form of the flow equations.

Here, in this work, our aim is to propose a method so that the structure and fluid component

is coupled by interface conditions not by imposing the force term

some key approaches related to our proposal :

eXtended Finite Element Method (XFEM):

Coupling Strategies (partitioned approach)

The extended finite element method (XFEM) is frequently used in order to incorporate discontinuous solution properties into the approximation space. Discontinuities inside elements, as they e.g. occur in two-phase/free-surface flows with implicit interface descriptions, can thereby be accounted for appropriately. In the extended finite element method (X-FEM), a standard space based finite element approximation is enriched by additional (special) functions using the framework of partition of unity. In other word XFEM is a numerical method, based on the Finite Element Method (FEM), that is especially designed for treating discontinuities.

In the X-FEM, the finite element mesh need not conform to the internal boundaries

Former researcher

multi-fluid (Chessa and Belytschko 2003; Coppola-Owen and Codina 2005; Fries and Belytschko 2010; Henning and Thomas-Peter 2011; Sven and Arnold 2007)

FSI

(Motasoares et al. 2006; Sawada and Tezuka 2011)

Coupling of multi-physics problems for instance FSI can be simulated in two approaches

monolithic approach

- Both the flow equations and structural equations are solved simultaneously

partitioned approach

- Requires a code developed for this particular combination of physical problems

- Flow equations and structural equations are solved separately

- Preserves software modularity

- Are typically explicit and staggered

- Advantage: allows solving the flow equations and the structural equations with

different , possibly more efficient techniques which have been developed specifically

for either flow equations or structural equations

What is our proposed approach

Here we proposed an Embedded FSI technique for Thin Deformable Structures using FEM with enrichment method based on XFEM

what is new in our work is that likes all non-conforming mesh methods, capable handle large deformations/motion of the structure because fluid and solid meshes are generated independently from each other with this difference that dislike immersed boundary and fictitious methods, the interaction of the fluid and the solid is taken into account through common interface conditions not by applying a force term

Important points of this method are as follows

• non-conforming mesh method can reduce human and computational costs and difficulties of the mesh

generation

• This method has a potential capability of handling the geometrical complexity of FSI problems, most of which

is caused by large deformation and translation or motion of structures.

• Partitioned approach for Coupling fluid-structure, allows solving the flow equations and the structural

equations with different, possibly more efficient techniques which have been developed specifically for either

flow equations or structural equations.

**OBJECTIVES**

The objective is to analyze Fluid deformable Structure Interaction with large structural deformations/motion without re-meshing process

To achieve the objective, we propose a non-conforming mesh method based on eXtended Finite Element Method (X-FEM) that is especially designed for treating discontinuities in velocity and pressure in fluid arises from structure. This method can reduce human and computational costs and difficulties of the mesh generation. At the end we will compare the results obtained by our method and the structure-conforming mesh methods.

**METHODOLOGY**

**Fluid**

**Structure**

**XFEM**

**FSI**

Governing equation of fluid

conservation of momentum

Mass conservation

If we assume Newtonian behavior, through the constitutive equation it is possible to relate the fluid stress tensor to the fluid velocity

Therefore the Navier-Stokes equations is

Boundary conditions

Governing equation of structure

Where:

boundary conditions

- Partitioned approach is used

Fluid Structure Interaction

- The influence of the structure on the fluid

- The influence of the fluid on the structure

eXtended Finite Element Method

- The Extended Finite Element Method (XFEM) is a numerical method, based on the Finite Element Method (FEM), that is especially designed for treating discontinuities

- Discontinuities are generally divided in strong and weak discontinuities.

- Strong discontinuities are discontinuities in the solution variable of a problem

- Weak discontinuities are discontinuities in the derivatives of the solution variable

In this study, two methods related to XFEM are used

An enrichment method using additional degree of freedom

A modified XFEM method in which no additional degrees of freedom are incorporated.

consider a piecewise linear approximation field

Enrichment method using additional degree of freedom

defined in a cut triangular element

the element can then be split into two sub-elements

and

weak discontinuous

strong discontinuous

Modified XFEM (without additional degree of freedom)

In this case no additional degrees of freedom are incorporated but the shape functions are modified so as to capture discontinuities

NA(A) = 1 NB(A) = 0 NC(A) = 0

NA(B) = 0 NB(B) = 1 NC(B) = 0

NA(C) = 0 NB(C) = 0 NC(C) = 1

NA(P+) = 1 NB(P+) = 0 NC(P+) = 0

NA(P-) = 0 NB(P-) = 1 NC(P-) = 0

NA(Q+) = 1 NB(Q+) = 0 NC(Q+) = 0

NA(Q-) = 0 NB(Q-) = 0 NC(Q-) = 1

Values at the vertices of the sub-triangles

PRELIMINARY RESULTS

Heat transfer problem

Stokes problem

We consider a heat transfer problem so that one thin fissure is embedded into a heat flow domain

The model equation for transient heat transfer by conduction

boundary conditions:

weak form of transient heat transfer eguation

where:

In finite element, the temperature is approached by the discretized form:

Matrix form of weak equation:

where:

The system has to be integrated in time, therefore we have

Enrichment method

For the elements which are intersected by the interface, approximation function consists of a standard finite element (FE) part and the enrichments parts

To define weak equation we decompose the discrete problem by using test functions from the linear, weak discontinuous and jump part, respectively

Matrix form:

where

Now, using the fact that the enrichment functions are local to each element, we eliminate

and

the elementary level before final assembly as follows

Modified XFEM method:

In this case for those elements that are cut by the interface, the classical conforming space, locally modified to accommodating discontinuities at the (given) interface

Note:

This method does not introduce any additional degrees of freedom

Results:

We consider the heat flow domain (0,6)×(0,6) with one thin fissure

Finite element space with conforming mesh

Enrichment method

Modified XFEM

Contour lines of Temperature

Deformable Structures:

An important field of FSI problems involves single or multiple thin and deformable structures like membranes

Extremely light.

The geometry depends strongly on the given stress distribution

Increasing lightness and slenderness brings along a higher susceptibility to flow-induced deformations and vibrations when exposed to the fluid load

In civil engineering this exemplifies in wind-induced effects on thin shells and membrane structures.

These wind effects can define the decisive design loads and therefore require an in-depth analysis. This phenomenon is called aeroelasticity in cases of an interaction between structure and wind. It can occur on constructions such as towers, high-rise buildings, bridges, cable and membrane roofs, etc

What we will to introduce:

Here, our project aim is to analyze Fluid deformable Structure Interaction with large structural deformations/motion without re-meshing process

The Stokes problem in differential form:

Weak formulation of Stokes problem

Where

Discrete problem for Galerkin Method

Matrix form of discrete problem

Consistently stabilized methods for the Stokes equations: (GLS)

Galerkin terms

Stabilization terms

Stabilization terms

residual of momentum equation

test function terms

It is then easy to see that the discrete problem is equivalent to a family of linear algebraic systems of the form

Where

Enrichment method

,

,

,

,

,

,

we decompose the discrete problem by using test functions from the standard and enrichment parts

Matrix for of discrete problem

Re-write the matrix of discrete problem

Static condensation

Modified XFEM method:

In this case for those elements that are cut by the interface, the classical conforming space, locally modified to accommodating discontinuities at the (given) interface

Note:

This method does not introduce any additional degrees of freedom

Results:

We consider the fluid domain (0,6)×(0,6) with one thin structure

TIMETABLE

showing contour lines of Pressure

showing contour lines of Velocity in X direction

showing contour lines of Velocity in Y direction

Courtesy of Daniel Baumgartner and Johannes Wolf

Conforming mesh

Enrichment Method

Modified XFEM

Conforming mesh

Enrichment Method

Modified XFEM

Conforming mesh

Enrichment Method

Modified XFEM

The material time derivative becomes a simple partial derivative with respect to time, such that

Structural acceleration

Cauchy stress tensor

Structural density

The structure is described using a Lagrangian description

Structural body forces